If a problem cant be investigated or not enough resources to investigate, then the PMgr needs to have a process to define those kind of problems and closes off the problems_________________John Hardesty
ITSM Manager's Certificate (Red Badge)

If they are low, then what is really the value in pursuing them if you have a workaround? Now, if you still have incidents occurring against these problems, then I would suggest using that as a means to raise the priority to raise visibility.

But we are talking RCA, which should be urgent for 95% of all problems because the root cause leads you to determining solutions.

We don't have this where I am, all problems are fairly urgent currently, so we get to root cause quickly, we also fix them quickly, a little too quickly.

This is a delicate issue. If you have a problem starving months and months with nobody taking care of it… it probably means that solving it is not so critical. You should review your stack of problems on a regular basis (for example once a month) and then close all the tickets starving since a too long time. Keeping them in the stack of the open problems will only overload problem management till making almost impossible to work when too many problems are open.

If you think all the problems open should remain open even if they are staving since a long time… it means you need more resources.

If you think you solve problems too quickly (as Titan said in his answer) pay attention… you are loosing money! (too much resources)

John says “this is problem management” I’d rather say this is an example of problem management. (and btw a very well described and detailed example, you understand ITIL concepts much faster when you can figure out concepts, so thanks for the effort)

In John’s example the problem is clearly identified, the team has resources to investigate and they find the root cause.

It happens (sometimes… most of the times… it depends on your organisation) that you don’t have enough resources to deal with all problems as you would. And so you have to act and respecting priorities cut off the “dead branches” to concentrate on the most important stuff. (as somehow John pointed out in his first reply)

Respecting ITIL the main activities of problem management are:

Problem control (handling Problems in an efficient and effective way)
Error control (to be aware of errors, to monitor them and to eliminate them when feasible and cost-justifiable)
Proactive prevention of Problems (identifying and resolving Problems before Incidents occur)

In my opinion taking such critical choices like closing an unresolved problem record is one of the most delicate and important parts of problem management. Sometimes more critical that bring a problem resolution till the happy end.